In the latter formula the separation of the 

 quotient into real and imaginary parts has 

 been effected by the usual and important 

 device of rationalizing the denominator, that 

 is, both numerator and denominator are multi- 

 plied by the complex conjugate of the de- 

 nominator or a;, - iy, • 

 Substitution of 



ajj = r^ cos 6.^, y^ = r^ sin 6^, 



^2 ^ ^2 ^°^ ^2' ^2 °° ^2 ^^^ ^2 

 gives, after some trigonometric substitutions, 



Figure 19 — The addition of two complex 



numbers z and z . 



_i =_i [cos (0j - d^) + i sin((9^ - $2)]. 

 ^2 '"2 



Thus the modulus of the product is the product of the moduli of the factors; and the amplitude 

 of the product is naturally obtained as the sum of the amplitudes. Similarly, the modulus of 

 the quotient is the quotient of the moduli; whereas the amplitude of the quotient may be taken 

 to be the difference between the amplitudes of numerator and denominator. This convention 

 as to amplitudes of product and quotient will be retained throughout. An example is illustrated 

 in Figure 20. 



Multiplication of a complex number by 

 i merely increases its amplitude by n-/2, or 

 rotates the representative vector on the dia- 

 gram counterclockwise through 77/2. Multipli- 

 cation by —i decreases the amplitude by n/2 

 and rotates the vector clockwise. 



The following formulas may be noted: 



zz* = {x + iy){x - iy) = ar^ + y^ = r^; 



z + z* = 2x, 2 - 3* = 2zy; 

 \z"\ = \z\" for real n, l^jSjl = l^il kal' 



|2./2„l = |3,l/|2-,l 



Figure 20 - Illustrating the product and 

 quotient of 2 and z . 



35 



